Another way to put this, the regular (lm) regression indicates that the more friends a student has, the happier they are, but looking closer this is not the case in all schools (and is, in fact, the opposite in some). These are considered fixed because they take on a predetermined set of values. SAS/SPSS uses Satterthwaite approximations. Thus, we will compare whether the addition of the random slopes to the random-intercepts model results in an improved fit. Finally, I highly recommend the textbook chapter on multilevel models in Cohen, Cohen, West and Aiken (2013) and the entire textbook by Gelman and Hill (2006). Instead of one general random-effect that captures how each observation deviates from the predicted fixed-effects, there will be multiple random-effects that capture how observations deviate within a cluster, and how each cluster deviates from the overall group. We examine the intra-class correlation (ICC) to determine if multi-level modeling is the correct choice for our analysis. Most of the examples found in the Mplus section can be run using the demo version. It is a positive correlation as well, but only poor schools (those with more students) had a positive correlation. install.packages(“texreg”) # Allows you to save tables. Not properly controlling for these differences, which you may often not know are there, will increase your chance of Type I error. Thus, while this 25 ms difference was not significant using OLS regression, it is significant in a MLM. In our example, this would indicate variability in the time participants took to begin speaking in English, but a constant relationship between the response language and the time. Chapter 9: Multilevel modeling with complex survey data view examples Description. By controlling for the variability in the time to speak due to each subject’s different baselines (i.e., random-intercepts), the remaining variability indicated a significant fixed-effect of language across individuals. She is interested in whether a matching or mismatching response language will affect the time it takes to begin the utterance. In addition, there is also the fixed-effect regression (solid line) that captures the overall group effect. Instead, observations from the same individual are likely to be “clustered,” and a MLM adds some extra parameters that control for this clustering. For instance, individuals may be nested within workgroups, or repeated measures may be nested within individuals. We wondered if happiness could be predicted by number of friends and/or GPA. A different intercept is estimated for each participant (dotted lines), assuming the same slope for all participants. Linear models and linear mixed effects models in R with linguistic applications. Cambridge, England: Cambridge University Press. In ordinary least squares (OLS) regression, we would model this data using the following formula: For each case, the “Time” score can be separated into fixed and random effects. Now that we specified our random effects, we can test whether the fixed effect of language is significant. Tutorials in Quantitative Methods for Psychology, 8(1): 52-69. For example, since we only used three color words out of the entire population of possible color words, we might use the items as another random-effect to control for “by-item” variability. In this case, Level 1 is the students. From this simple example, we can easily scale up our model. Data analysis using regression and multilevel/hierarchical models. Baayen et al. Either are acceptable. Peugh, J.L. Our ICC is greater than 0, meaning we were correct to think of this as a multi-level problem. The goal of multi-level modeling is to draw a conclusion about the general sample that you have while controlling for differences you are not trying to explain (in this example, rich vs. poor). (2011) provide excellent in-depth discussions. To assess whether the addition of the random-slopes improved the fit of our model, we can use a goodness of fit test known as a likelihood ratio (LR) chi-square difference test (i.e., nested model test; Snijders & Bosker, 2012). Although mathematically sophisticated, MLMs are easy to use once familiar with some basic concepts. If we knew this information, we would put SES as another Level 2 predictor. In the left-hand portion of the figure below, we can see a graphical representation of this random-intercepts model. We may also have reason to believe that the relationship between time to speak and the response language might differ for each individual, in addition to the baseline differences. (2008). An example Mumper, M. (2017, January). Thus, we will create a model with only an intercept as a fixed effect, and compare it to our random-intercepts model. (2006). The fixed effects include the intercept (B0) and the slope (B1) for the dichotomous independent variable "Language." Applied multiple regression/correlation analysis for the behavioral sciences (3rd Ed.). Journal of Experimental Psychology: General, 143(5), 2020-2045. Beyond Multiple Linear Regression: Applied Generalized Linear Models and Multilevel Models in R (R Core Team 2020) is intended to be accessible to undergraduate students who have successfully completed a regression course through, for example, a textbook like Stat2 (Cannon et al. Notice how the correlation between Friends and Happiness has changed when you collapse across all rich and poor schools. This will be important. That brings our assumed R=3 down to about 1.2. The expression (B0 + B1 * Language|Subject) tells the model to estimate different intercepts and slopes for each individual, as shown in the right-hand portion of the figure below. Multilevel models (MLMs, also known as linear mixed models, hierarchical linear models or mixed-effect models) have become increasingly popular in psychology for analyzing data with repeated measurements or data organized in nested levels (e.g., students in classrooms). SO, Let’s try running this as a multilevel, random effects model: Using a multi-level model allows us to separate the within-group effects from the between-group effects, whereas regular regression blends them together into a single coefficient. The result is significant ((X(1) = 3.96, p < .05), suggesting that the addition of the fixed-effect of language improves model fit. The Quantitative Methods for Psychology, 10(1): 13- 28. (2008) and West et al. Uh-oh. Westfall, J., Kenny, D. A., & Judd, C. M. (2014). The issue with this approach is that we repeatedly sampled from the same individual, and this violates the OLS assumption that observations are independent from each other. Baayen et al. If you were trying to generalize your findings or use them to argue for/show a need for an intervention, your results would be misleading and could cause problems. (2006). Great, our correlation between Friends and Happiness is -.76 and we have a correlation between GPA and Happiness of -.05. Therefore, students who received initial instruction in SEM with lavaan should have little di culty using other (commercial) SEM programs in the future. In the present example, our MLM gave us a more accurate interpretation - that no main effect of Friends existed for all schools generally. you will need to refit the models with the package lmerTest installed and loaded. The ICC measures the degree of clustering in our data and answers the question, “How much does my Level 2 predict the total variance of my study?” If your ICC is greater than 0, you have a multi-level study. We find an intercept of 522.11 and a slope of -24.89, suggesting that participants took about 522 ms to begin speaking when responding in English, but began speaking 25 ms faster when responding in Spanish. NOTE: Although I am telling you that some schools are rich and some are poor, you may not know this information when you come in and try to do the analysis. Multilevel data occur when observations are nested within groups, for example, when students are nested within schools in a district. We can include more predictors at both the micro-level and macro-level, which can be used as random slopes, and can even interact (a cross-level interaction). Winter, B. Retrieved from http://arxiv.org/pdf/1308.5499.pdf (PDF, 2.69MB). We estimate the variability for each random-effect and use that to control for the variance when estimating the significance our fixed-effects. An introduction to hierarchical linear modeling. Multilevel analyses are applied to data that have some form of a nested structure. That’s a huge reduction, but 1.2 is still bigger than 1: it means that the epidemic will double every 3 weeks. We can also have nested random effects. Our model would then include a (B0│Subject)+(B0 |Item) for random intercepts by-subject and by-item. Baayen, R.H., Davidson, D.J., & Bates, D.M. COVID-19 resources for psychologists, health-care workers and the public. Modeling Longitudinal Data by Robert Weiss; Logistic Regression and Related Methods Applied Logistic Regression (Second Edition) by David Hosmer and Stanley Lemeshow; An Introduction to Categorical Data Analysis by Alan Agresti; Multilevel Modeling Introduction to Multilevel Modeling by … Introductions to using MLMs are available for R (Winters, 2013), SPSS (Hayes, 2006), and SAS (Peugh, 2010). Statistical power and optimal design in experiments in which samples of participants respond to samples of stimuli. To do so, we create a base model that is identical to the original model minus the effect of interest. A practical guide to multilevel modeling. In this primer, I use a simple example to demonstrate the fundamental principles of MLMs. What could bring R below 1, short of a full-scale lockdown? Thus, I highly encourage you to continue learning about MLMs. Baayen, R.H., Davidson, D.J., & Bates, D.M. Our regression equation now looks like: Timei = B0 + B1 * Languagei + (B0|Subject) + ei. Multilevel modelling. (2013). The color words are always presented in English. Also, I am only using six schools to demonstrate, but in a more “normal” case of nested design there may be way too many schools to treat SES as a fixed factor in the design. Let’s test if these formulas work with an example from each set of schools (one rich and one poor). Base Model: Timei = B0 + (B0|Subject) +ei, Random Intercepts: Timei = B0 + B1* Languagei + (B0|Subject) +ei. In other words, this formula indicates that there are multiple responses per subject, and that each response will depend on that subject’s baseline. PSA is the monthly e-newsletter of the APA Science Directorate. This flexibility makes MLM an attractive option for researchers. Call for Papers/Proposals/Nominations (4), © 2021 American Psychological Association. One solution is to estimate degrees of freedom to get p-values. Thus, the random-intercept model appears to be the best fit. This combination of different “levels” of analysis gives rise to the term multi-level modeling. This is accomplished using an LR test too. (2011) provide excellent in-depth discussions. Now we have a positive, significant effect of Friends and no effect of GPA. arXiv:1308.5499. Rich schools have r = -.76 and poor schools have r = .77. However, in our mixed (lmer) regression, Friends had a larger (2.16), but non-significant effect. Returning to our example, we are going to add a random-effect for “Subject” that characterizes each participant’s idiosyncratic variation from the fixed-effect estimates. Let’s try running a normal regression on all the data at once. Now let’s see what happens if you run a normal regression on rich and poor schools separately. This is known as a random-intercepts model, because we are going to estimate a different intercept for each subject. Thus, we can model our data at the observation level (micro-level) and at the cluster level (macro-level). The regular regression did not reflect what was happening in each school type. If you want to report main effects and interactions instead of slopes, you might convert your mixed model into an ANOVA. In our regular (lm) regression, Friends had a significant effect, b = 1.40 (p < .001). Our model will control for this within-cluster variance and test the fixed-effect estimates against the remaining between-cluster variance. ---
title: "Chapter 16: Multilevel Modeling"
author: "Teresa G. Borowski"
output:
  html_document:
    theme: cerulean
    highlight: textmate
    fontsize: 8pt
    toc: true
    number_sections: true
    code_download: true
    toc_float:
      collapsed: false
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(message = FALSE)
knitr::opts_chunk$set(warning = FALSE)
knitr::opts_chunk$set(fig.width=5)
knitr::opts_chunk$set(fig.height=3.75)
knitr::opts_chunk$set(fig.align='center') 
```

# Prepping your R

##Packages you will need:

- install.packages("lme4") # Allows you to fit linear mixed-effects models.
- install.packages("effects") # Helps you visualize interactions
- install.packages("corrplot") # Helps you to visualize correlations 
- install.packages("stargazer") # Makes tables
- install.packages("car") # Helps with Diagnostics
- install.packages("MASS") # For stepwise regression
- install.packages("ggplot2") # To create plots
- install.packages("texreg") # Allows you to save tables.

##Load the libraries you need.

```{r}
library(lme4)
library(effects)
library(corrplot)
library(stargazer)
library(car)
library(MASS)
library(ggplot2)
library(texreg)
```


# The Data Set:

This is a special type of data where there is only one measurement per student (i.e., there is no within subject variable), and students are nested in schools so we must control for the effects of schools. 

Multilevel data occur when observations are nested within groups, for example, when students are nested within schools in a district.

Our simple story -

We looked at 6 schools (3 rich and 3 poor) with 40 students in each rich school and 160 students in each poor school, and we measured them on Happiness, number of Friends, and GPA. We wondered if happiness could be predicted by number of friends and/or GPA.

Our Level 1 will be the lowest unit of observation - those nested in groups. In this case, Level 1 is the students.

Our Level 2 refers to the groups in which the level 1 units are nested; in this case Level 2 is the schools.

## Simulating the Dataset:

### Make your equations
Create a distribution for each school, making the variance of rich schools small and the variance of poor schools large. We've also made the equations different so that our X (number of friends) coefficient is negative in rich schools but positive in the poor schools. We keep our Z (GPA) coefficient the same for all schools. 

```{r}
set.seed(42)
nrich=40
npoor=160
#Paramaters
S.F.Rich=-2 # setting paramaeters for the Friends variable in the Rich schools
S.F.Poor=6 # setting parameters for the Friends variable in the Poor schools

S.G.Rich=.7 # setting parameters for the GPA variable in the Rich schools
S.G.Poor=.7 # setting parameters for the GPA variable in the Poor schools
# Rich Schools
# School 1
X1 <- rnorm(nrich, 10, 2) # number of friends
Z1 <- runif(nrich, 1.0, 4.0) # GPA
Y1 <- S.F.Rich*X1 + S.G.Rich*Z1 + 80 + rnorm(nrich, sd= 5) # Our equation to create Y

# School 2
X2 <- rnorm(nrich, 10, 2) # number of friends
Z2 <- runif(nrich, 1.0, 4.0) # GPA
Y2 <- S.F.Rich*X2 + S.G.Rich*Z2 + 75 + rnorm(nrich, sd= 5)

# School 3
X3 <- rnorm(nrich, 10, 2) # number of friends
Z3 <- runif(nrich, 1.0, 4.0) # GPA
Y3 <- S.F.Rich*X3 + S.G.Rich*Z3 +90 + rnorm(nrich, sd= 5)

# Poor Schools
# School 4
X4 <- rnorm(npoor, 5, 2) #number of friends
Z4 <- runif(npoor, 1.0, 4.0) #GPA
Y4 <- S.F.Poor*X4 + S.G.Poor*Z4 + 35 + rnorm(npoor, sd = 10)

# School 5
X5 <- rnorm(npoor, 5, 2) #number of friends
Z5 <- runif(npoor, 1.0, 4.0) #GPA
Y5 <- S.F.Poor*X5 + S.G.Poor*Z5 + 40 + rnorm(npoor, sd = 10)

# School 6
X6 <- rnorm(npoor, 5, 2) #number of friends
Z6 <- runif(npoor, 1.0, 4.0) #GPA
Y6 <- S.F.Poor*X6 + S.G.Poor*Z6 + 50 + rnorm(npoor, sd = 10)

``` 

### Make your data frames
Now that we have our equations, we're going to create the data frame for each school.

```{r}
# The 3 Rich Schools:
Student.Data.School.1<-data.frame(Happiness=Y1, Friends=X1, GPA=Z1)
Student.Data.School.2<-data.frame(Happiness=Y2, Friends=X2, GPA=Z2)
Student.Data.School.3<-data.frame(Happiness=Y3, Friends=X3, GPA=Z3)

# The 3 Poor Schools:
Student.Data.School.4<-data.frame(Happiness=Y4, Friends=X4, GPA=Z4)
Student.Data.School.5<-data.frame(Happiness=Y5, Friends=X5, GPA=Z5)
Student.Data.School.6<-data.frame(Happiness=Y6, Friends=X6, GPA=Z6)

```

### Testing the formulas
Let's test if these formulas work with an example from each set of schools (one rich and one poor).

```{r}
# Rich - School 1
corr.student = cor(Student.Data.School.1)
corrplot(corr.student, method = "number",diag = FALSE,type = "lower")
```

Great, our correlation between Friends and Happiness is -.76 and we have a correlation between GPA and Happiness of -.05.

```{r}
# Poor - School 4
corr.student = cor(Student.Data.School.4)
corrplot(corr.student, method = "number",diag = FALSE,type = "lower")
```

Great, our correlation between Friends and Happiness is .77, which is higher AND in the opposite direction than in rich schools. There was no correlation between GPA and Happiness.

Pay attention to the fact that the schools have different correlation values for Friends and Happiness depending on if they're rich or poor. Rich schools have r = -.76 and poor schools have r = .77. For GPA and Happiness, only the rich school had a correlation. This will be important. 

NOTE: Although I am telling you that some schools are rich and some are poor, you may not know this information when you come in and try to do the analysis. If we knew this information, we would put SES as another Level 2 predictor. 


### Put 'em together and what have you got!?
Now that we know they work, let's put all schools together into one dataset. 

```{r}
All.Schools.Data <- rbind(Student.Data.School.1, Student.Data.School.2, Student.Data.School.3, Student.Data.School.4, Student.Data.School.5, Student.Data.School.6) 
head(All.Schools.Data)
```

### Adding Student and School Variables
Now let's add in Student ID's and the Schools as variables.

```{r}
# Adding the subject variable (Student ID)
All.Schools.Data$StudentID<-seq(1:nrow(All.Schools.Data))
# Did it work?
head(All.Schools.Data)
# Yes!

# Adding the School variable. 
All.Schools.Data$School<-c(rep(1, nrich), rep(2,nrich), rep(3, nrich), rep(4, npoor), 
                           rep(5, npoor), rep(6, npoor))
```

### Test the full dataset
Let's see what the scatter plots for each school.

- Schools 1-3 are richand schools 4-6 are poor.


# Plotting 

Now let's plot up raw data.

First, let's plot with Friends as our IV.

```{r}
theme_set(theme_bw(base_size = 12, base_family = "")) 

# Friends
Model.Plot.Friends <-ggplot(data = All.Schools.Data, aes(x = Friends, y=Happiness,group=School))+	
  facet_grid( ~ School)+	
  geom_point(aes(colour = School))+	
  geom_smooth(method = "lm", se = TRUE, aes(colour = School))+	
  xlab("Friends")+ylab("Happiness")+	
  theme(legend.position = "none")	
Model.Plot.Friends	
```

Now, let's plot with GPA as our IV.

```{r}
Model.Plot.GPA <-ggplot(data = All.Schools.Data, aes(x =GPA, y=Happiness,group=School))+	
  facet_grid( ~ School)+	
  geom_point(aes(colour = School))+	
  geom_smooth(method = "lm", se = TRUE, aes(colour = School))+	
  xlab("GPA")+ylab("Happiness")+	
  theme(legend.position = "none")	
Model.Plot.GPA	
```

Let's see what our correlations look like when lumping all the rich and poor schools together.

```{r}
corr.student = cor(All.Schools.Data[,1:3])
corrplot(corr.student, method = "number",diag = FALSE,type = "lower")
```

Notice how the correlation between Friends and Happiness has changed when you collapse across all rich and poor schools. The correlation is now .24 (which is between Rich and Poor schools). It is a positive correlation as well, but only poor schools (those with more students) had a positive correlation.
Also notice that the correlation between GPA and Happiness is .05, which is the same as for the rich school - the only one that had showed a correlation. 


# Regular Regression

## All Data
Let's try running a normal regression on all the data at once.

First, let's center the variables.

```{r}
All.Schools.Data$Friends.C<-scale(All.Schools.Data$Friends, scale = FALSE)[,]
All.Schools.Data$GPA.C<-scale(All.Schools.Data$GPA, scale = FALSE)[,]
```

Now we can run the regression.

```{r}
Reg.Model<-lm(Happiness ~ Friends.C + GPA.C, data = All.Schools.Data)
summary(Reg.Model)
#plot(Reg.Model)
```

Putting all schools together in a regular regression, we have a positive, significant effect of Friends and no effect of GPA. But what happens when we separate the schools by rich or poor status?

## By School
Now let's see what happens if you run a normal regression on rich and poor schools separately.

```{r}
# Rich Schools
School.Rich.Reg.Model<-lm(Happiness ~ Friends.C + GPA.C, data = subset(All.Schools.Data, School<4))
summary(School.Rich.Reg.Model)
```

Notice, we have a negative, significant effect for Friends but no effect for GPA.

```{r}
# Poor Schools
School.Poor.Reg.Model<-lm(Happiness ~ Friends.C + GPA.C, data = subset(All.Schools.Data, School>3))
summary(School.Poor.Reg.Model)
```

Uh-oh. Now we have a positive, significant effect of Friends and no effect of GPA.

## What's the problem?

Remember! Although I am telling you that some schools are rich and some are poor, you may not know this information when you come in and try to do the analysis. If we knew this information, we could put SES as another Level 2 predictor. Also, I am only using six schools to demonstrate, but in a more "normal" case of nested design there may be way too many schools to treat SES as a fixed factor in the design.

The regular regression did not reflect what was happening in each school type. It gave us a positive effect (Friends) when only poor schools had a positive effect. Collapsing across school types in this case was not ideal because different things were happening within each school type, compromising the generalizability of the findings.

Another way to put this, the regular (lm) regression indicates that the more friends a student has, the happier they are, but looking closer this is not the case in all schools (and is, in fact, the opposite in some). If you were trying to generalize your findings or use them to argue for/show a need for an intervention, your results would be misleading and could cause problems. 

SO,
Let's try running this as a multilevel, random effects model:


# Random Effects Model

Using a multi-level model allows us to separate the within-group effects from the between-group effects, whereas regular regression blends them together into a single coefficient. 


## Running the Random Effects Model 


### First, we run the null model.

```{r}
Null<-lmer(Happiness ~ 1 # This simply means Happiness predicted by the intercept
                  +(1|School), # each school gets its own intercept 
                  data=All.Schools.Data, REML = FALSE)
summary(Null)
```

We examine the intra-class correlation (ICC) to determine if multi-level modeling is the correct choice for our analysis. The ICC measures the degree of clustering in our data and answers the question, "How much does my Level 2 predict the total variance of my study?" If your ICC is greater than 0, you have a multi-level study. 

```{r}
ICC.Model<-function(Model.Name) {
  tau.Null<-as.numeric(lapply(summary(Model.Name)$varcor, diag))
  sigma.Null <- as.numeric(attr(summary(Model.Name)$varcor, "sc")^2)
  ICC.Null <- tau.Null/(tau.Null+sigma.Null)
  return(ICC.Null)
}

ICC.Model(Null)
```

Our ICC is greater than 0, meaning we were correct to think of this as a multi-level problem.


### Next, we run the Level 1 predictor (GPA).

```{r}
The.Model.1<-lmer(Happiness ~ GPA.C 
                  +(1|School),
                data=All.Schools.Data, REML = FALSE)
summary(The.Model.1)
```

The results indicate that a student's GPA does not have an effect on their Happiness when controlling for Level 2 fluctuations in Happiness. 


### Finally, we run the Level 2 predictor (Friends).

This model allows the variable Friends to vary between schools.


```{r}
The.Model.2<-lmer(Happiness ~ Friends.C + GPA.C 
                  +(1+Friends.C|School), # each school gets its own intercept, and Friends can vary as a function of the school.
                data=All.Schools.Data, REML = FALSE)
summary(The.Model.2)
```

The results indicate that the number of Friends a student has does not have an effect on Happiness when controlling for the random effects of Level 2 influences.


### What do the results mean?

In our regular (lm) regression, Friends had a significant effect, b = 1.40 (p < .001). However, in our mixed (lmer) regression, Friends had a larger (2.16), but non-significant effect.

Why is this important? The goal of multi-level modeling is to draw a conclusion about the general sample that you have while controlling for differences you are not trying to explain (in this example, rich vs. poor). Not properly controlling for these differences, which you may often not know are there, will increase your chance of Type I error. Because the effect of Friends was different in different schools, it makes sense that the multi-level model (MLM) did not show a significant effect. In the present example, our MLM gave us a more accurate interpretation - that no main effect of Friends existed for all schools generally. 


## Checking your Assumptions

For a comprehensive look at how to run diagnostics, please see [Chapter 12](http://ademos.people.uic.edu/Chapter12.html) and [Chapter 18](http://ademos.people.uic.edu/Chapter18.html)  


# P-Values

*P*-values are hotly disputed in Mixed Models. 

One solution is to estimate degrees of freedom to get p-values. SAS/SPSS uses Satterthwaite approximations. There are also Kenward-Roger approximations (see Westfall, Kenny, & Judd, 2014). Either are acceptable. Making a number of assumptions, these outputs will mirror the results of a traditional ANOVA fairly closely. 

install.packages("lmerTest")

This is a mixed model add-on package to calculate p-values based on a certain set of assumptions.
Each ddf is a different method of attaining p-values, so you can choose which to run. I give you three examples below. 
you will need to refit the models with the package `lmerTest` installed and loaded.

```{r}
library(lmerTest)
The.Model.2<-lmer(Happiness ~ Friends.C + GPA.C 
                  +(1+Friends.C|School),
                data=All.Schools.Data, REML = FALSE)

summary(The.Model.2,ddf = "lme4")
summary(The.Model.2, ddf = "Satterthwaite") # SAS method
```

If you want to report main effects and interactions instead of slopes, you might convert your mixed model into an ANOVA.

```{r}
anova(The.Model.2,ddf = "Kenward-Roger") # Kenny et al suggested method
```

# References

Huta, V. (2014). When to use hierarchical linear modeling. The Quantitative Methods for Psychology, 10(1): 13- 28.

Westfall, J., Kenny, D. A., & Judd, C. M. (2014). Statistical power and optimal design in experiments in which samples of participants respond to samples of stimuli. Journal of Experimental Psychology: General, 143(5), 2020-2045.

Woltman, H., Feldstain, A., MacKay, J. C., & Rocchi, M. (2012). An introduction to hierarchical linear modeling. Tutorials in Quantitative Methods for Psychology, 8(1): 52-69.


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, A Language, not a Letter: Learning Statistics in R. install.packages(“lme4”) # Allows you to fit linear mixed-effects models.
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